# Name of the set $\{x : x_1 \leq x_2 \leq \cdots \leq x_n\}$?

I was looking for a standard name of the set $\{x \in A : x_1 \leq x_2 \leq \cdots \leq x_n\}$, where $A = [0,1]^n$ or $A = [0,\infty)^n$. I think I saw this recently, but now I cannot find it anywhere.

Thanks in advance.

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@Kannappan: No, I mean that each coordinate is restricted to be bigger than the previous one. – Set Master Mar 30 '12 at 16:46
You could call it the set of all (weakly) increasing, or non-decreasing, $n$-tuples from $[0,1]$ or $[0,\infty)$. – joriki Mar 30 '12 at 16:50
@Kannappan: The $x_i$ are the components of $x$. – joriki Mar 30 '12 at 16:50
The best thing, I think, is not to introduce a name for such a thing (unless you are about to start writing a complete book about it...) – Mariano Suárez-Alvarez Mar 30 '12 at 17:29
In the case $A=[0,1]^n$ this is an $n$-simplex. – Grumpy Parsnip Mar 30 '12 at 19:04

## 3 Answers

The second one is a Weyl chamber, the first one a Weyl chamber (for the same root system) truncated at level 1.

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Odds are, unless the context cries for this nomenclature, using it will not clarify anything —and I am pretty sure that if the term ringed a bell for the intended audience and/or the OP then the question would not have been asked :) – Mariano Suárez-Alvarez Mar 30 '12 at 18:14
@Didier: Thanks! This is the name I couldn't remember. :) – Set Master Mar 30 '12 at 18:30
@Mariano: Bingo! :-) – Did Mar 30 '12 at 19:48
@SetMaster: You are welcome. – Did Mar 30 '12 at 19:49

In geometric terms this is an n-simplex called a Schläfli orthoscheme when $A = [0,1]^n$.

The case when $A = [0,\infty)^n$ is an unbounded convex region, but very similar to a simplex in that it is simply a limit of the simplex as coordinates of the first figure are scaled up. One may well refer to this figure as the non-negative cone of the respective $(n-1)$-simplex.

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If someone started talking to me about an «$n$-simplex with an orthogonal corner» I would surely ask what she means... – Mariano Suárez-Alvarez Mar 30 '12 at 17:30
I thought the simplex was defined by the sum of the coordinates being 1, not by the coordinates being nondecreasing. – Did Mar 30 '12 at 17:32
@Didier: A regular n-simplex can be construed as a sum of n+1 coordinates that add up to 1. However this simplex is not regular in the sense of having all faces and angles congruent. – hardmath Mar 30 '12 at 18:08
@MarianoSuárez-Alvarez: In my rush to improve, I went wrong! New term, new link. – hardmath Mar 30 '12 at 19:16

it is the weyl chamber for the symmetric group

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Could you provide some more information? – draks ... Apr 3 '12 at 10:26